Title: A Theoretical and Computational Journey with Elliptic Curves and
Modular Forms.
Abstract: Elliptic curves and modular forms are very important objects in
number theory. Although they're relatively classical, they're still at the
center of much research; the most famous example being Fermat's Last
Theorem following from Taniyama-Shimura conjecture-that-is-now-a-theorem:
"All Elliptic Curves are Modular" (via works of Wiles and others). In this
talk we attempt to understand some of the theoretical and computational
aspects of the statement above, with an emphasis on how mathematical
software can be used not just in computations, but also as a tool to help
understand some complicated theoretical notions. The number theoretic
background required for the talk is minimal; in particular, all objects
introduced will be defined and motivated.